Describe the theta format.
Syntax
help theta
help thss
Description
theta is a packed matrix containing information about both a model structure and its nominal or estimated parameters. It also contains other relevant information about the identification result.
This model format is the basic format with the System Identification Toolbox. It is used by all parametric identification methods and it can be transformed to many other model representations. See the tables in the beginning of this chapter for more details.
The internal format of the theta format is intended to be transparent to the user. The basic way to display the information is to use the present command. Some specific information is retrieved from the format by the functions getmfth, getncap, gett, and th2par. This entry gives the details of the internal representation, but this information is not necessary for most users of the System Identification Toolbox. The formats differ whether the underlying model is in state-space form or of the input-output black box character.
I. For the general multi-input single-output linear model structure
A, B, C, D, and F are polynomials in the delay operator of orders na, nb, nc, nd, and nf, respectively. If the system has nu inputs, nb, nf and nk are row vectors of dimension nu containing information about the orders and delays associated with each of the inputs. In the case of a time series (no u), B and F are not defined.
Let n be the sum of all the orders (the number of estimated parameters) and let
r = max(n, 7, 6 + 3 nu )
Then theta is a (3+n) r matrix organized as follows:
- Row 1 has entries: estimated variance of e, sampling interval T, nu, na, nb, nc, nd, nf, and nk.
Row 2 has entries: FPE (Akaike's Final Prediction Error), year, month, date, hour, minute and command by which the model was generated. The matrix entry (2,7) thus contains the coded number of which command generated the model. This number is less than 20 for the black-box models of the type above.
Row 3 is the vector of estimated parameters, A, B ,C, D, and F, (excluding leading 1s and 0s).
Rows 4 to 3+n contain the estimated covariance matrix.
- For continuous-time models, a possible element (4+n,1) contains a dead-time for the system. This is used when computing frequency functions in
th2ff.
II. For models that are defined as state-space structures there is an underlying M-file that defines the structure. This M-file is called by
[A,B,C,D,K,X0] = mfname(par,T,aux)
(See mf2th.) For model structures that are defined by ms2th, the name of this M-file is ssmodx9 or ssmodx8 and the argument aux is the actual model structure ms created by modstruc. Notice in particular that multi-output ARX models are internally represented as state-space models with the aid of ssmodx9.
Suppose that the number of estimated parameters is n and that the length of the name of your M-file mfname is r. Suppose also that the model has ny outputs and that the size of the argument aux above is nr times nc. Then theta is a matrix of dimension
max(n, ny, nc, 7 + r) by 3+ n + nr + ny)
organized as follows:
- Row 1 has entries: determinant of innovations covariance, sampling interval, number of inputs, number of outputs, number of estimated parameters, number of rows of
aux, number of columns of aux, and name of
M-file that defines the structure.
- Row 2 contains the entries: FPE (Akaike's Final Prediction Error), year, month, date, hour, minute, and command by which the model was generated. This last one is a number larger than 20 for state-space structures.
- Entry
(2,8) is interpreted as follows: "1" means that the underlying parameterization is in continuous time, using ssmodx9. "ll" means that the underlying parameterization is in continuous time, to be sampled by first-order-hold, using ssmodx8. "2" means that it is in discrete time, again using ssmodx9. `3" means that the model is a multivariate ARX model. "4" means that the underlying parameterization is in discrete time and user defined. "5" means that the model is a user-defined continuous-time parameterization, equipped with sampling inhibition when called with a negative value of T.
- Row 3 contains the estimated (nominal) values of the parameters.
- Rows 4 to 3 + n contain the estimated covariance matrix of the parameters.
- Rows 4 + n to 3 + n + nr contain the matrix
aux.
- Rows 4 +n + nr to 3+n + nr + ny contain the covariance matrix of the innovations of the model.
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